How does a hula hoop master gravity? Mathematicians prove that shape matters

(Press-News.org) Hula hooping is so commonplace that we may overlook some interesting questions it raises: “What keeps a hula hoop up against gravity?” and “Are some body types better for hula hooping than others?” A team of mathematicians explored and answered these questions with findings that also point to new ways to better harness energy and improve robotic positioners. 

The results are the first to explain the physics and mathematics of hula hooping.

“We were specifically interested in what kinds of body motions and shapes could successfully hold the hoop up and what physical requirements and restrictions are involved,” explains Leif Ristroph, an associate professor at New York University’s Courant Institute of Mathematical Sciences and the senior author of the paper, which appears in the Proceedings of the National Academy of Sciences.

To answer these questions, the researchers replicated, in miniature, hula hooping in NYU’s Applied Mathematics Laboratory. They tested different shapes and motions in a series of experiments on robotic hula hoopers using 3D-printed bodies of different shapes (e.g., cylinders, cones, hourglass shapes) to represent human forms at one-tenth the size. These shapes were driven to gyrate by a motor, replicating the motions we take when hula hooping. Hoops approximately 6 inches in diameter were launched on these bodies, with high-speed video capturing the movements.

The results showed that the exact form of the gyration motion or the cross-section shape of the body (circle versus ellipse) wasn’t a factor in hula hooping. 

“In all cases, good twirling motions of the hoop around the body could be set up without any special effort,” Ristroph explains. 

However, keeping a hoop elevated against gravity for a significant period of time was more difficult, requiring a special “body type”—one with a sloping surface as “hips” to provide the proper angle for pushing up the hoop and a curvy form as a “waist” to hold the hoop in place. 

“People come in many different body types—some who have these slope and curvature traits in their hips and waist and some who don’t,” notes Ristroph. “Our results might explain why some people are natural hoopers and others seem to have to work extra hard.”

The paper’s authors conducted mathematical modeling of these dynamics to derive formulas that explained the results—calculations that could be used for other purposes.

“We were surprised that an activity as popular, fun, and healthy as hula hooping wasn’t understood even at a basic physics level,” says Ristroph. “As we made progress on the research, we realized that the math and physics involved are very subtle, and the knowledge gained could be useful in inspiring engineering innovations, harvesting energy from vibrations, and improving in robotic positioners and movers used in industrial processing and manufacturing.”

The paper’s other authors were Olivia Pomerenk, an NYU doctoral student, and Xintong Zhu, an NYU undergraduate at the time of the study.

The work was supported by a grant from the National Science Foundation (DMS-1847955). 

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